More Things To Read


My long streak of posting something every day will end. There’s just no keeping up mathematics content like this indefinitely, not with my stamina. But it’s not over just quite yet. I wanted to share some stuff that people had brought to my attention and that’s just too interesting to pass up.

The first comes from … I’m not really sure. I lost my note about wherever it did come from. It’s from the Continuous Everywhere But Differentiable Nowhere blog. It’s about teaching the Crossed Chord Theorem. It’s one I had forgotten about, if I heard it in the first place. The result is one of those small, neat things and it’s fun to work through how and why it might be true.

Next comes from a comment by Gerry on a rather old article, “What’s The Worst Way To Pack?” Gerry located a conversation on MathOverflow.net that’s about finding low-density packings of discs, circles, on the plane. As these sorts of discussions go, it gets into some questions about just what we mean by packings, and whether Wikipedia has typos. This is normal for discovering new mathematics. We have to spend time pinning down just what we mean to talk about. Then we can maybe figure out what we’re saying.

And the last I picked up from Elke Stangl, of what’s now known as the elkemental Force blog. She had pointed me first to lecture notes from Dr Scott Aaronson which try to explain quantum mechanics from starting principles. Normally, almost invariably, they’re taught in historical sequence. Aaronson here skips all the history to look at what mathematical structures make quantum mechanics make sense. It’s not for casual readers, I’m afraid. It assumes you’re comfortable with things like linear transformations and p-norms. But if you are, then it’s a great overview. I figure to read it over several more times myself.

Those notes are from a class in Quantum Computing. I haven’t had nearly the time to read them all. But the second lecture in the series is on Set Theory. That’s not quite friendly to a lay audience, but it is friendlier, at least.

Packing For Higher Dimensions


You may have heard of the sphere-packing problem. If you haven’t, let me brief you. It’s a problem about how to pack a bunch of spheres. Particularly, it’s about how to place spheres, all the same size, so there’s as little wasted space as possible.

It’s not an easy problem. Johannes Kepler, whom you remember as the astronomer with the gold nose because you’ve mixed him up with Tycho Brahe, studied it. He conjectured, in 1611, that the best packing you could do was the “close packing”. You know this pattern because it’s what a stack of oranges ends up being. We believe he was right. A computer-assisted proof was published in 2005.

But if we’re comfortable with mathematics we know a sphere, or a ball, doesn’t have to be something as boring as the balls we have in the real world. We could consider a circle to be a two-dimensional sphere. We could make something four-dimensional that looks a lot like a sphere. Or five-dimensional. Or 800-dimensional, if we have some reason to do this. (We do!) And optimization problems can be strange things. How many dimensions of space something has can affect how easy or hard a problem is. But just having more dimensions doesn’t mean the problem is harder. Sometimes having a vaster space means the problem becomes easier.

There’s recently been a breakthrough in the eight dimension. A paper by Maryna S Viazovska, with the Berlin Mathematical School and the Humboldt University of Berlin, seems to have worked out the densest possible packing for eight-dimensional spheres. And better, it ties into this beautiful pattern known as the E8 lattice. The MathsByAGirl blog recently posted an essay about that, and I’d like to recommend folks over there.

And, because I’m like this, I’d like to point folks over to one of my old essays. I’d got to wondering what the least efficient sphere packings were. The answers might surprise you.

Infinite Circle Packings


Here’s an engaging and apparently new mathematics blog. Euclidean Brew-Space — the name is explained in its Nomenclature Derivation page — here discusses the problem of tiling a space. As a bonus it includes a neat trick with a pint glass.

Euclidean Brew-Space

Question of the Week:
Before I get started with this week’s entry, I wanted to share a neat trick with you!

So like the video says, if you can figure out how I knew when to stop adding height to the glass, please comment below!


Alright, so this week I want to discuss a problem that I’m sure we have all encountered in some way, shape or form: how can we pack the most bottles of beer in a limited amount of space? If you don’t imbibe, that’s fine – this problem does extend to cans of soda or any round item you might want to shove in your refrigerator. As a homebrewer, I have a completely ludicrous number of empty bottles just waiting to be filled with fresh beer and since I made the transition to kegging my brews, the problem has become a serious storage issue. How can…

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The Math Blog Statistics, April 2014


Another month’s gone by and the statistics about viewership were pretty gratifying for April 2014. But I’m feeling awfully good about the place, because I’ve felt more gratified by the mathematics blog lately. It’s felt to me like there’ve been more comments and more interaction the past couple weeks, and it’s felt like it’s getting closer to supporting a community, which is thrilling, if not exactly measurable given what WordPress shares with me.

In March 2014, according to last month’s statistics survey, there were 453 views from 257 distinct viewers. That jumped pretty noticeably this month to 565 views, albeit from 235 distinct viewers, a views-per-visitor jump from 1.76 to 2.40. I suspect there’s some archive-bingers, and I’m happy to give anyone that thrill. It’s my greatest viewer count since June 2013, and the fourth-highest since December 2012 when WordPress started sharing statistics on unique visitors. I also noted at the start of April that while I’d reached 14,000 visitors in March I’d need a stroke of luck to reach 15,000 in April. I came close: the month topped out with my 14,931st view.

The most popular articles of the past thirty days were:

  1. How Dirac Made Every Number, the answer to that puzzle of how to construct any counting number using precisely four 2’s and ordinary operations (it’s a forehead slapper once you’ve seen it)
  2. Reading the Comics, April 27, 2014: The Poetry of Calculus Edition, as everyone wants to see some calculus poetry
  3. Can You Be As Clever As Dirac For A Little Bit in which that Dirac puzzle was laid out and the rules given
  4. How Many Trapezoids I Can Draw, always with the trapezoids
  5. Reading the Comics, April 21, 2014: Bill Amend In Name Only Edition, which includes a bundle of a lot of mathematics comics from Bill Amend’s FoxTrot in case you need some

The countries sending me the most viewers were the United States (294), Canada (65), Denmark (29), Austria (27), and the United kingdom (26), and I count nine countries sending me at least ten views each, which I think is a record but I haven’t been keeping track of that number. Sending me a single viewer each were Belgium, Brazil, Ecuador, Finland, Greece, Hungary, Malaysia, Morocco, Oman, Sweden, and Venezuela. Belgium, Brazil, Hungary, and Sweden were single-country viewers last month, and Hungary’s got a three-month single-viewer streak going. So, ah, hi, whoever that is in Hungary. Apparently nobody has ever visited me from Honduras.

Once again there’s a shortage of search term poetry, but there were some fair queries the past month, including:

January 2014’s Statistics


So how does the first month of 2014 compare to the last month of 2013, in terms of popularity? The raw numbers are looking up: I went from 176 unique visitors looking at 352 pages in December up to 283 unique visitors looking at 498 pages. If WordPress’s statistics are to be believed that’s my greatest number of page views since June of 2013, and the greatest number of visitors since February. This hurt the ratio of views per visitor a little, which dropped from 2.00 to 1.76, but we can’t have everything unless I write stuff that lots of people want to read and they figure they want to read a lot more based on that, which is just crazy talk. The most popular articles, though, were:

  1. Something Neat About Triangles, this delightful thing about forming an equilateral triangle starting from any old triangle.
  2. How Many Trapezoids I Can Draw, with my best guess for how many different kinds of trapezoids there are (and despite its popularity I haven’t seen a kind not listed here, which surprises me).
  3. Factor Finding, linking over to IvaSallay’s quite interesting blog with a great recreational mathematics puzzle (or educational puzzle, depending on how you came into it) that drove me and a friend crazy with this week’s puzzles.
  4. What’s The Worst Way To Pack? in which I go looking for the least-efficient packing of spheres and show off these neat Mystery Science Theater 3000 foam balls I got.
  5. Reading The Comics, December 29, 2013, the old year’s last bunch of mathematics-themed comic strips.

The countries sending me readers the most often were the United States (281), Canada (52), the United Kingdom (25), and Austria (23). Sending me just a single reader each this past month were a pretty good list:
Bulgaria, France, Greece, Israel, Morocco, the Netherlands, Norway, Portugal, Romania, Russia, Serbia, Singapore, South Korea, Spain, and Viet Nam. Returning on that list from last month are Norway, Romania, Spain, and Viet Nam, and none of those were single-country viewers back in November 2013.

What’s The Worst Way To Pack?


While reading that biography of Donald Coxeter that brought up that lovely triangle theorem, I ran across some mentions of the sphere-packing problem. That’s the treatment of a problem anyone who’s had a stack of oranges or golf balls has independently discovered: how can you arrange balls, all the same size (oranges are near enough), so as to have the least amount of wasted space between balls? It’s a mathematics problem with a lot of applications, both the obvious ones of arranging orange or golf-ball shipments, and less obvious ones such as sending error-free messages. You can recast the problem of sending a message so it’s understood even despite errors in coding, transmitting, receiving, or decoding, as one of packing equal-size balls around one another.

A collection of Mystery Science Theater 3000 foam balls which I got as packing material when I ordered some DVDs.

The “packing density” is the term used to say how much of a volume of space can be filled with balls of equal size using some pattern or other. Patterns called the cubic close packing or the hexagonal close packing are the best that can be done with periodic packings, ones that repeat some base pattern over and over; they fill a touch over 74 percent of the available space with balls. If you don’t want to follow the Mathworld links before, just get a tub of balls, or crate of oranges, or some foam Mystery Science Theater 3000 logo balls as packing materials when you order the new DVD set, and play around with a while and you’ll likely rediscover them. If you’re willing to give up that repetition you can get up to nearly 78 percent. Finding these efficient packings is known as the Kepler conjecture, and yes, it’s that Kepler, and it did take a couple centuries to show that these were the most efficient packings.

While thinking about that I wondered: what’s the least efficient way to pack balls? The obvious answer is to start with a container the size of the universe, and then put no balls in it, for a packing fraction of zero percent. This seems to fall outside the spirit of the question, though; it’s at least implicit in wondering the least efficient way to pack balls to suppose that there’s at least one ball that exists.

Continue reading “What’s The Worst Way To Pack?”